CAREER: K-stability and moduli spaces of higher dimensional varieties

职业：K-稳定性和高维簇的模空间

• 批准号: 2237139
• 负责人: Yuchen Liu
• 金额: $ 46.38万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-07-15 至 2028-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2237139&HistoricalAwards=false
• 关键词: CAREER; stability; moduli; spaces; higher

Algebraic varieties are geometric shapes arising from solutions to polynomial equations. A central problem in the study of algebraic varieties, known as the moduli problem, is to find a classifying space of algebraic varieties. One prominent strategy to solve the moduli problem is to study canonical geometric structures on algebraic varieties, known as Einstein metrics. This project is to study the moduli problem for algebraic varieties with the help of Einstein metrics. The problems are foundational and have deep connections to other areas of mathematics, such as commutative algebra, differential geometry, and mathematical physics. The project includes research and learning opportunities for students and early-career researchers in algebraic geometry, through seminars, workshops, summer schools, and other activities.More specifically, the PI will investigate K-stability and K-moduli spaces of Fano varieties, connections to other well-studied moduli spaces, and related problems for singularities. Firstly, the PI will construct compact moduli spaces of K-unstable Fano varieties through Kaehler-Ricci solitons. Secondly, the PI will study explicit moduli spaces of Fano threefolds and log Fano pairs via the framework of wall crossings, and establish connections to moduli spaces of curves and K3 surfaces. Thirdly, The PI will construct moduli spaces of log Calabi-Yau pairs connecting K-moduli spaces and KSBA moduli spaces. Finally, the PI will study boundedness and moduli problems for singularities using the theory of normalized volumes.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

代数品种是由溶液到多项式方程式产生的几何形状。代数品种研究中的一个核心问题（称为模量问题）是找到代数品种的分类空间。解决模量问题的一种突出策略是研究代数品种（称为爱因斯坦指标）的规范几何结构。该项目是在爱因斯坦指标的帮助下研究代数品种的模量问题。这些问题是基本的，并且与其他数学领域有着深厚的联系，例如交换代数，差异几何和数学物理学。该项目包括通过研讨会，研讨会，暑期学校和其他活动为学生和代数几何学的早期研究人员进行研究和学习机会。更具体地说，PI将研究Fano品种的K稳定性和K-Moduli空间，与其他良好的Moduli空间的连接，以及奇数问题的其他相关问题。首先，PI将通过Kaehler-Icci solitons构建K- Unstable Fano品种的紧凑型模量空间。其次，PI将通过壁桥的框架研究Fano三倍的Fano三倍的显式模量空间，并建立与曲线和K3表面模量空间的连接。第三，PI将构造连接K-Moduli空间和KSBA模量空间的Log Calabi-yau对的模量空间。最后，PI将使用归一化量理论研究界限的有限性和模量问题。该奖项反映了NSF的法定任务，并使用基金会的知识分子优点和更广泛的影响审查标准，被视为值得通过评估来提供支持。